1.2 Functions from the Graphical Viewpoint
(This topic is also in Section 1.2 in Finite Mathematics, Applied Calculus and Finite Mathematics and Applied Calculus)
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Let us start by looking at the definition in the textbook (and also in the Chapter Summary)
Graph of a Function
The graph of the function f is the set of all points (x, f(x)) in the xyplane, where we restrict the values of x to lie in the domain of f.
To obtain the graph of a function, plot points of the form (x, f(x)) for several values of x in the domain of f. The shape of the entire graph can usually be inferred from sufficiently many points.
Example
To sketch the graph of the function
f(x) = x^{2}   Function notation 
y = x^{2}   Equation notation 
with domain the set of all real numbers, first choose some values of x in the domain and compute the corresponding ycoordinates.
x  3  2  1  0  1  2  3 
y = x^{2}  9  4  1  0  1  4  9 
Plotting these points gives the picture on the left, suggesting the graph on the right

If only the graph of a function is given to begin with, we say that the function has been specified graphically. Here is an example of a graphically specified function.
The following graph shows the total population in state and federal prisons in 19701997 as a function of time in years (t = 0 represents 1970).^{*}
* Data are approximate. Sources: Bureau of Justice Statistics, New York State Dept. of Correctional Services/The New York Times, January 9, 2000, p. WK3.
The next example shows that plotting a few points might not convey anough infomration to enable us to draw a graph.
Here is what we get if we carefully plot the points we just obtained.
The Graph of a Piecewise Defined Function
Since piecewisedefined functions are based on more than oneformula, their graphs are composed of more than one curve. Here is Example 4 in the textbook: Let
f(x)  = 

1  if  4 x < 1  x  if  1 x 1  x^{2}1  if^{ }  1 < x 2^{ } 


The graph of f consists of portions of three graphs superimposed. To see how they fit together, click the buttons under the graph of f below.
Now try some of the exercises in Section 1.2 of the textbook, or press "Review Exercises" on the sidebar to see a collection of exercises that covers the whole of Chapter 1.
Last Updated: March, 2006
Copyright © 2001, 2007 Stefan Waner